1. The problem ( Primal problem ) is equivalent to

2. The problem ( Dual problem ) is equivalent to

Note that The rest follows from the definitions of the problems ( Primal problem ) and ( Dual problem ).

(Necessary and sufficient optimality conditions).The vectors form a solution of the problem ( Primal problem ) and a geometric multiplier pair if and only if the following four conditions hold.

If the form a solution of the problem ( Primal problem ) and a geometric multiplier pair then the statements ( Primal feasibility ) and ( Dual feasibility ) follow from the definitions and ( Lagrangian optimality ),( Complementary slackness ) follow from the proposition ( Geometric multiplier property ).

Conversely, using the conditions of the theorem we obtain The the equality follows from the propositions ( Visualization lemma ) and ( Weak duality theorem ).